TPTP Problem File: ITP132^2.p
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%------------------------------------------------------------------------------
% File : ITP132^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Number_Partition problem prob_191__5326608_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Number_Partition/prob_191__5326608_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 362 ( 96 unt; 62 typ; 0 def)
% Number of atoms : 893 ( 283 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3364 ( 66 ~; 26 |; 46 &;2788 @)
% ( 0 <=>; 438 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 130 ( 130 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 61 usr; 4 con; 0-6 aty)
% Number of variables : 981 ( 64 ^; 850 !; 9 ?; 981 :)
% ( 58 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:28:18.421
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (60)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere1490568538miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s1003936772cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups1340683514dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
number2016821345itions: ( nat > nat ) > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_p,type,
p: nat > nat ).
% Relevant facts (253)
thf(fact_0_partitions,axiom,
number2016821345itions @ p @ n ).
% partitions
thf(fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq @ nat @ k @ n ).
% \<open>k \<le> n\<close>
thf(fact_2_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atMost @ A @ X )
= ( set_ord_atMost @ A @ Y ) )
= ( X = Y ) ) ) ).
% atMost_eq_iff
thf(fact_3_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062,axiom,
( ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
= ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
@ ( set_ord_atMost @ nat @ n ) ) ) ).
% \<open>(\<Sum>i\<le>n - k. (p(k := p k - 1)) i * i) = (\<Sum>i\<le>n. (p(k := p k - 1)) i * i)\<close>
thf(fact_5_calculation,axiom,
( ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
= ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( p @ I ) @ I )
@ ( set_ord_atMost @ nat @ n ) )
@ k ) ) ).
% calculation
thf(fact_6_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,A3: set @ B] :
( ( groups1340683514dd_sum @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_7_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,A3: set @ A,G: C > B,B3: set @ C] :
( ( times_times @ B @ ( groups1340683514dd_sum @ A @ B @ F @ A3 ) @ ( groups1340683514dd_sum @ C @ B @ G @ B3 ) )
= ( groups1340683514dd_sum @ A @ B
@ ^ [I: A] :
( groups1340683514dd_sum @ C @ B
@ ^ [J: C] : ( times_times @ B @ ( F @ I ) @ ( G @ J ) )
@ B3 )
@ A3 ) ) ) ).
% sum_product
thf(fact_8_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R: A,F: B > A,A3: set @ B] :
( ( times_times @ A @ R @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) )
= ( groups1340683514dd_sum @ B @ A
@ ^ [N: B] : ( times_times @ A @ R @ ( F @ N ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_9_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,A3: set @ B,R: A] :
( ( times_times @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
= ( groups1340683514dd_sum @ B @ A
@ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_10_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_11_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_12_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ! [X4: A,K3: A] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) )
| ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_13_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_14_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I2 @ K ) ) ) ).
% atMost_iff
thf(fact_15_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ N2 )
=> ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_16_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_17_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_18_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_19_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ J2 @ K )
=> ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_20_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_21_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_22_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
| ( ord_less_eq @ nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_23_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B4 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_24_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_25_partitions__bounds,axiom,
! [P2: nat > nat,N2: nat,I2: nat] :
( ( number2016821345itions @ P2 @ N2 )
=> ( ord_less_eq @ nat @ ( P2 @ I2 ) @ N2 ) ) ).
% partitions_bounds
thf(fact_26_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_27_le__diff__iff_H,axiom,
! [A4: nat,C2: nat,B4: nat] :
( ( ord_less_eq @ nat @ A4 @ C2 )
=> ( ( ord_less_eq @ nat @ B4 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A4 ) @ ( minus_minus @ nat @ C2 @ B4 ) )
= ( ord_less_eq @ nat @ B4 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_28_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_29_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_30_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_31_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_32_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_33_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_34_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_35_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_36_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_37_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_38_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
= N2 ) ).
% nat_mult_1_right
thf(fact_39_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_40_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere216010020id_add @ A )
=> ! [K4: set @ B,F: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K4 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups1340683514dd_sum @ B @ A @ F @ K4 ) @ ( groups1340683514dd_sum @ B @ A @ G @ K4 ) ) ) ) ).
% sum_mono
thf(fact_41_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U ) ) ) ) ) ).
% atMost_def
thf(fact_42_sum__subtractf__nat,axiom,
! [A: $tType,A3: set @ A,G: A > nat,F: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1340683514dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( minus_minus @ nat @ ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) @ ( groups1340683514dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_43_partitions__parts__bounded,axiom,
! [P2: nat > nat,N2: nat] :
( ( number2016821345itions @ P2 @ N2 )
=> ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ P2 @ ( set_ord_atMost @ nat @ N2 ) ) @ N2 ) ) ).
% partitions_parts_bounded
thf(fact_44_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_sum_Oreindex__bij__witness,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,I2: C > B,J2: B > C,T: set @ C,H: C > A,G: B > A] :
( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( member @ C @ ( J2 @ A5 ) @ T ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T )
=> ( ( J2 @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T )
=> ( member @ B @ ( I2 @ B5 ) @ S ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ S )
= ( groups1340683514dd_sum @ C @ A @ H @ T ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_50_sum_Oeq__general__inverses,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ B,K: B > C,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
( ! [Y2: B] :
( ( member @ B @ Y2 @ B3 )
=> ( ( member @ C @ ( K @ Y2 ) @ A3 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( member @ B @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
= ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_51_sum_Oeq__general,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ B,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
( ! [Y2: B] :
( ( member @ B @ Y2 @ B3 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A3 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: C] :
( ( ( member @ C @ Ya @ A3 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ( ( member @ B @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
= ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% sum.eq_general
thf(fact_52_sum_Ocong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A,H: B > A] :
( ( A3 = B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
= ( groups1340683514dd_sum @ B @ A @ H @ B3 ) ) ) ) ) ).
% sum.cong
thf(fact_53_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_54_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B3: set @ C,A3: set @ B] :
( ( groups1340683514dd_sum @ B @ A
@ ^ [I: B] : ( groups1340683514dd_sum @ C @ A @ ( G @ I ) @ B3 )
@ A3 )
= ( groups1340683514dd_sum @ C @ A
@ ^ [J: C] :
( groups1340683514dd_sum @ B @ A
@ ^ [I: B] : ( G @ I @ J )
@ A3 )
@ B3 ) ) ) ).
% sum.swap
thf(fact_55_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ! [X4: A,K3: A] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) )
& ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_56_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.right_neutral
thf(fact_57_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% mult.left_neutral
thf(fact_58_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
= ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% fun_upd_upd
thf(fact_59_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_60_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F2: B > A,X2: B,Y4: A,Z2: B] : ( if @ A @ ( Z2 = X2 ) @ Y4 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_61_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_62_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_63_partitions__def,axiom,
( number2016821345itions
= ( ^ [P3: nat > nat,N: nat] :
( ! [I: nat] :
( ( ( P3 @ I )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I )
& ( ord_less_eq @ nat @ I @ N ) ) )
& ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( P3 @ I ) @ I )
@ ( set_ord_atMost @ nat @ N ) )
= N ) ) ) ) ).
% partitions_def
thf(fact_64_partitionsI,axiom,
! [P2: nat > nat,N2: nat] :
( ! [I3: nat] :
( ( ( P2 @ I3 )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I3 )
& ( ord_less_eq @ nat @ I3 @ N2 ) ) )
=> ( ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( P2 @ I ) @ I )
@ ( set_ord_atMost @ nat @ N2 ) )
= N2 )
=> ( number2016821345itions @ P2 @ N2 ) ) ) ).
% partitionsI
thf(fact_65_partitionsE,axiom,
! [P2: nat > nat,N2: nat] :
( ( number2016821345itions @ P2 @ N2 )
=> ~ ( ! [I4: nat] :
( ( ( P2 @ I4 )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I4 )
& ( ord_less_eq @ nat @ I4 @ N2 ) ) )
=> ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( P2 @ I ) @ I )
@ ( set_ord_atMost @ nat @ N2 ) )
!= N2 ) ) ) ).
% partitionsE
thf(fact_66_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_67_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_68_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_69_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A4: A,B4: A] :
( ( ( times_times @ A @ A4 @ B4 )
= ( zero_zero @ A ) )
= ( ( A4
= ( zero_zero @ A ) )
| ( B4
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_70_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B4 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B4 ) ) ) ) ).
% mult_cancel_left
thf(fact_71_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A4: A,C2: A,B4: A] :
( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B4 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B4 ) ) ) ) ).
% mult_cancel_right
thf(fact_72_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_73_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_0_right
thf(fact_74_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_75_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_zero
thf(fact_76_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_77_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A4 ) ).
% bot_nat_0.extremum
thf(fact_78_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_79_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_80_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_81_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_82_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_83_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_84_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_85_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B] :
( ( groups1340683514dd_sum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A3 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_86_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B4 ) )
= ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_87_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [A4: A,C2: A] :
( ( ( times_times @ A @ A4 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_88_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C2: A,B4: A] :
( ( C2
= ( times_times @ A @ B4 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_89_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C2: A,A4: A] :
( ( ( times_times @ A @ C2 @ A4 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_90_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C2: A,B4: A] :
( ( C2
= ( times_times @ A @ C2 @ B4 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_91_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_92_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_93_gr0,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ) ).
% gr0
thf(fact_94_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_95_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_96_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A,B4: A] :
( ( ( times_times @ A @ A4 @ B4 )
!= ( zero_zero @ A ) )
=> ( ( A4
!= ( zero_zero @ A ) )
& ( B4
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_97_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A4: A,B4: A] :
( ( ( times_times @ A @ A4 @ B4 )
= ( zero_zero @ A ) )
=> ( ( A4
= ( zero_zero @ A ) )
| ( B4
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_98_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A4: A,B4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ B4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_99_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% mult_left_cancel
thf(fact_100_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B4 @ C2 ) )
= ( A4 = B4 ) ) ) ) ).
% mult_right_cancel
thf(fact_101_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B6: A] :
( ( minus_minus @ A @ A6 @ B6 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_102_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_103_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_104_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_105_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
!= ( zero_zero @ A ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_106_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq @ nat @ A4 @ ( zero_zero @ nat ) )
=> ( A4
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_107_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq @ nat @ A4 @ ( zero_zero @ nat ) )
= ( A4
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_108_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_109_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_110_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M )
= ( zero_zero @ nat ) )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_111_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_112_mult__0,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere1490568538miring @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_114_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_115_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B4 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_116_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_117_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_118_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_119_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B4: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_120_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_121_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_122_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_123_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% mult_left_mono
thf(fact_124_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_125_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_126_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,B4: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% split_mult_pos_le
thf(fact_127_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ A4 ) ) ) ).
% zero_le_square
thf(fact_128_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B4: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_129_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B4: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_130_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_131_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_132_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_133_partitions__zero,axiom,
! [P2: nat > nat] :
( ( number2016821345itions @ P2 @ ( zero_zero @ nat ) )
= ( P2
= ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% partitions_zero
thf(fact_134_mult__left__le,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [C2: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ A4 ) ) ) ) ).
% mult_left_le
thf(fact_135_mult__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_136_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_137_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_138_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A3: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_139_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A3: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) ) ) ) ).
% sum_nonneg
thf(fact_140_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times @ nat @ M @ N2 ) )
=> ( ( N2
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_141_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_142_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_143_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_144_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_145_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_146_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_147_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_148_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_149_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_150_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_151_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_152_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_153_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_154_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_155_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_156_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_157_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_158_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_159_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_161_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_162_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_163_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_164_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_165_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_166_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_167_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_168_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_169_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A6: A,B6: A] : ( times_times @ A @ B6 @ A6 ) ) ) ) ).
% mult.commute
thf(fact_170_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( times_times @ A @ B4 @ ( times_times @ A @ A4 @ C2 ) )
= ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_171_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B4: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A4 = B4 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A4: A,C2: A,B4: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B4 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_173_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_174_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_175_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A4: A,C2: A,M: A > B,B4: B,D2: B] :
( ( A4 != C2 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A4 @ B4 ) @ C2 @ D2 )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D2 ) @ A4 @ B4 ) ) ) ).
% fun_upd_twist
thf(fact_176_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
( ( Z != X )
=> ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_177_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
( ( fun_upd @ B @ A @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_178_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
( ( ( F @ X )
= Y )
=> ( ( fun_upd @ B @ A @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_179_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= ( fun_upd @ A @ B @ G @ X @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_180_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A6: A,B6: B,X2: A] : ( if @ B @ ( X2 = A6 ) @ B6 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_181_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_182_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% diff_left_mono
thf(fact_183_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_184_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_185_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_186_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B4 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B4 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_187_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s1003936772cancel @ A )
=> ! [B4: A,C2: A,A4: A] :
( ( times_times @ A @ ( minus_minus @ A @ B4 @ C2 ) @ A4 )
= ( minus_minus @ A @ ( times_times @ A @ B4 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) ) ) ) ).
% left_diff_distrib'
thf(fact_188_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s1003936772cancel @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B4 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B4 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_189_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_190_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.comm_neutral
thf(fact_191_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_192_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_193_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_194_partitions__remove1__bounds,axiom,
! [P2: nat > nat,N2: nat,K: nat,I2: nat] :
( ( number2016821345itions @ P2 @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( P2 @ K ) )
=> ( ( ( fun_upd @ nat @ nat @ P2 @ K @ ( minus_minus @ nat @ ( P2 @ K ) @ ( one_one @ nat ) ) @ I2 )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I2 )
& ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% partitions_remove1_bounds
thf(fact_195_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_196_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_197_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_198_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_199_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_200_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_201_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_202_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B4 ) )
= ( ord_less @ A @ B4 @ A4 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_203_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_204_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_205_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_206_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_207_less__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_208_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_209_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_210_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( A4 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_212_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( A4 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_213_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_214_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_215_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B4 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_216_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P4: A > $o] :
? [X5: A] : ( P4 @ X5 ) )
= ( ^ [P5: A > $o] :
? [N: A] :
( ( P5 @ N )
& ! [M4: A] :
( ( ord_less @ A @ M4 @ N )
=> ~ ( P5 @ M4 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_217_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_218_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_219_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_220_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_221_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_222_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_223_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_224_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_225_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_226_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_227_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_228_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_229_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_230_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_231_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_232_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_233_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% less_asym'
thf(fact_234_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_235_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_236_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% dense
thf(fact_237_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% order.asym
thf(fact_238_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_239_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_240_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_241_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_242_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_243_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_244_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_245_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_246_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_247_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_248_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X3 ) )
& ~ ( P @ Y3 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_249_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y3: A] :
( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% measure_induct_rule
thf(fact_250_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_251_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_252_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
% Type constructors (43)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 )
=> ( minus @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s1003936772cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd,axiom,
dvd @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_8,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_10,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_13,axiom,
minus @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( p @ I ) @ I )
@ ( set_ord_atMost @ nat @ n ) )
@ k )
= ( minus_minus @ nat @ n @ k ) ) ).
%------------------------------------------------------------------------------